Reduction of symplectic principal $\mathbb{R}$-bundles

We describe a reduction process for symplectic principal $\mathbb{R}$-bundles in the presence of a momentum map. This type of structures plays an important role in the geometric formulation of non-autonomous Hamiltonian systems. We apply this procedure to the standard symplectic principal $\mathbb{R}$-bundle associated with a fibration $π:M\to\mathbb{R}$. When $π$ is a principal $G$-bundle and $G_ν$ denotes the isotropy group associated with an element $ν$ in the dual to the Lie algebra of $G$, we use the reduction process in order to describe a Poisson structure on the quotient manifold $M/G_ν$ whose symplectic leaves are isomorphic to the coadjoint orbit $\mathcal{O}_ν$ . Moreover, we show a reduction process for non-autonomous Hamiltonian systems on symplectic principal $\mathbb{R}$-bundles.

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